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Unit Roots

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Abstract

Models with autoregressive unit roots play a major role in modern time series analysis and are especially important in macroeconomics, where questions of shock persistence arise, and in finance, where martingale concepts figure prominently in the study of efficient markets. The literature on unit roots is vast and applications of unit root testing span the social, environmental and natural sciences. The present article overviews the theory and concepts that underpin this large field of research and traces the originating ideas and econometric methods that have become central to empirical practice.

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Bibliography

  • Andrews, D.W.K. 1993. Exactly median-unbiased estimation of first-order autoregressive/unit root models. Econometrica 61: 139–166.

    Article  Google Scholar 

  • Andrews, D.W.K., and P. Guggenberger. 2006. Asymptotics for stationary very nearly unit root processes. Mimeo: Yale University.

    Google Scholar 

  • Bai, J. 1997. Estimating multiple break one at a time. Econometric Theory 13: 315–352.

    Article  Google Scholar 

  • Bai, L., and P. Perron. 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66: 47–78.

    Article  Google Scholar 

  • Bai, L., and P. Perron. 2006. Multiple structural change models: A simulation analysis. In Econometric theory and practice, ed. Dean Corbae, B.E. Hansen, and S.N. Durlauf. New York: Cambridge University Press.

    Google Scholar 

  • Baltagi, B.H., and C. Kao. 2000. Nonstationary panels, cointegration in panels and dynamic panels: A survey. Advances in Econometrics 15: 7–51.

    Article  Google Scholar 

  • Banerjee, A., R. Lumsdaine, and J. Stock. 1992. Recursive and sequential tests of the unit root and trend break hypotheses: Theory and international evidence. Journal of Business and Economic Statistics 10: 271–287.

    Google Scholar 

  • Beare, B.K. 2006. Unit root testing with unstable volatility. Mimeo: Yale University.

    Google Scholar 

  • Berkes, I., and L. Horváth. 2006. Convergence of integral functionals of stochastic processes. Econometric Theory 22: 304–322.

    Article  Google Scholar 

  • Beveridge, S., and C.R. Nelson. 1981. A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’. Journal of Monetary Economics 7: 151–174.

    Article  Google Scholar 

  • Bhargava, A. 1986. On the theory of testing for unit roots in observed time series. Review of Economic Studies 52: 369–384.

    Article  Google Scholar 

  • Box, G.E.P. and Jenkins, G.M. 1976. Time series analysis: Forecasting and control, rev. edn. San Francisco: Holden Day.

    Google Scholar 

  • Byrne, J.P., and R. Perman. 2006. Unit roots and structural breaks: A survey of the literature. Mimeo: University of Strathclyde.

    Google Scholar 

  • Campbell, J.Y., and P. Perron. 1991. Pitfalls and opportunities: What macroeconomists should know about unit roots (with discussion). In NBER macroeconomics annual 1991, ed. O.J. Blanchard and S. Fischer. Cambridge, MA: MIT Press.

    Google Scholar 

  • Campbell, J.Y., and R.J. Shiller. 1988. Interpreting cointegrated models. Journal of Economic Dynamics and Control 12: 505–522.

    Article  Google Scholar 

  • Cavaliere, G. 2004. Unit root tests under time-varying variances. Econometric Reviews 23: 259–292.

    Article  Google Scholar 

  • Cavaliere, G., and A.M.R. Taylor. 2007. Testing for unit roots in time series models with non-stationary volatility. Journal of Econometrics 140(2): 919–947.

    Article  Google Scholar 

  • Chan, N.H., and C.Z. Wei. 1987. Asymptotic inference for nearly nonstationary AR (1) processes. Annals of Statistics 15: 1050–1063.

    Article  Google Scholar 

  • Chan, N.H., and C.Z. Wei. 1988. Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16: 367–401.

    Article  Google Scholar 

  • Choi, I. 2001. Unit roots for panel data. Journal of International Money and Finance 20: 249–272.

    Article  Google Scholar 

  • Choi, I., and P.C.B. Phillips. 1993. Testing for a unit root by frequency domain regression. Journal of Econometrics 59: 263–286.

    Article  Google Scholar 

  • Christiano, L.J. 1992. Searching for a break in GNP. Journal of Business & Economic Statistics 10: 237–250.

    Google Scholar 

  • de Jong, R. 2004. Addendum to ‘asymptotics for nonlinear transformations of integrated time series’. Econometric Theory 20: 623–635.

    Google Scholar 

  • DeJong, D.N., and C.H. Whiteman. 1991a. Reconsidering trends and random walks in macroeconomic time series. Journal of Monetary Economics 28: 221–254.

    Article  Google Scholar 

  • DeJong, D.N., and C.H. Whiteman. 1991b. The temporal stability of dividends and stock prices: evidence from the likelihood function. American Economic Review 81: 600–617.

    Google Scholar 

  • Dickey, D.A. 1976. Estimation and hypothesis testing in nonstationary time series. Ph.D. thesis, Iowa State University.

    Google Scholar 

  • Dickey, D.A., and W.A. Fuller. 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74: 427–431.

    Google Scholar 

  • Dickey, D.A., and W.A. Fuller. 1981. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49: 1057–1072.

    Article  Google Scholar 

  • Diebold, F.X., and M. Nerlove. 1990. Unit roots in economic time series. Advances in Econometrics 8: 3–70.

    Google Scholar 

  • Dolado, J.J., T. Jenkinson, and S. Sosvilla-Rivero. 1990. Cointegration and unit roots. Journal of Economic Surveys 4: 249–273.

    Article  Google Scholar 

  • Duffie, D. 1988. Security markets: Stochastic models. San Diego: Academic Press.

    Google Scholar 

  • Elliott, G., T.J. Rothenberg, and J.H. Stock. 1995. Efficient tests of an autoregressive unit root. Econometrica 64: 813–836.

    Article  Google Scholar 

  • Engle, R.F., and T. Bollerslev. 1986. Modeling the persistence of conditional variances. Econometric Reviews 5: 1–50.

    Article  Google Scholar 

  • Engle, R.F., and C.W.J. Granger. 1987. Co-integration and error correction: Representation, estimation, and testing. Econometrica 55: 251–276.

    Article  Google Scholar 

  • Evans, G.B.A., and N.E. Savin. 1981. Testing for unit roots: 1. Econometrica 49: 753–779.

    Article  Google Scholar 

  • Evans, G.B.A., and N.E. Savin. 1984. Testing for unit roots: 2. Econometrica 52: 1241–1269.

    Article  Google Scholar 

  • Fuller, W.A. 1976. Introduction of statistical time series. New York: Wiley.

    Google Scholar 

  • Ghysels, E., and D.R. Osborn. 2001. Econometric analysis of seasonal time series. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Giraitis, L., and P.C.B. Phillips. 2006. Uniform limit theory for stationary autoregression. Journal of Time Series Analysis 27: 51–60.

    Article  Google Scholar 

  • Gouriéroux, C., E. Renault, and N. Touzi. 2000. Calibration by simulation for small sample bias correction. In Simulation-based inference in econometrics: Methods and applications, ed. R.S. Mariano, T. Schuermann, and M. Weeks. Cambridge: Cambridge University Press.

    Google Scholar 

  • Gouriéroux, C., P.C.B. Phillips, and J. Yu. 2007. Indirect inference for dynamic panel models. Discussion Paper No. 1550, Cowles Foundation, Yale University.

    Google Scholar 

  • Hall, R.E. 1978. Stochastic implications of the life cycle-permanent income hypothesis. Journal of Political Economy 86: 971–987.

    Article  Google Scholar 

  • Hall, A. 1989. Testing for a unit root in the presence of moving average errors. Biometrika 76: 49–56.

    Article  Google Scholar 

  • Hall, P., and C.C. Heyde. 1980. Martingale limit theory and its application. New York: Academic Press.

    Google Scholar 

  • Harris, D., S. Leybourne, and B. McCabe. 2007. Modified KPSS tests for near integration. Econometric Theory 23: 355–363.

    Article  Google Scholar 

  • Hlouskova, J., and M. Wagner. 2006. The performance of panel unit root and stationarity tests: Results from a large scale simulation study. Econometric Reviews 25: 85–116.

    Article  Google Scholar 

  • Hong, S.H., and P.C.B. Phillips. 2005. Testing linearity in cointegrating relations with an application to purchasing power parity. Discussion Paper No. 1541, Cowles Foundation, Yale University.

    Google Scholar 

  • Hu, L., and P.C.B. Phillips. 2004. Dynamics of the federal funds target rate: a nonstationary discrete choice approach. Journal of Applied Econometrics 19: 851–867.

    Article  Google Scholar 

  • Hylleberg, S., R.F. Engle, C.W.J. Granger, and S. Yoo. 1990. Seasonal integration and cointegration. Journal of Econometrics 44: 215–238.

    Article  Google Scholar 

  • Ibragimov, R., and P.C.B. Phillips. 2007. Regression asymptotics using martingale convergence methods. Discussion Paper No. 1473, Cowles Foundation, Yale University.

    Google Scholar 

  • Johansen, S. 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12: 231–254.

    Article  Google Scholar 

  • Kapetanios, G. 2005. Unit root testing against the alternative hypothesis of up to m structural breaks. Journal of Time Series Analysis 26: 123–133.

    Article  Google Scholar 

  • Karlsen, H.A., T. Myklebust, and D. Tjøstheim. 2007. Nonparametric estimation in a nonlinear cointegration model. Annals of Statistics 35(1).

    Google Scholar 

  • Kasparis, I. 2004. Detection of functional form misspecification in cointegrating relations. Mimeo: University of Nottingham.

    Google Scholar 

  • Kim, J.Y. 1994. Bayesian asymptotic theory in a time series model with a possible nonstationary process. Econometric Theory 10: 764–773.

    Article  Google Scholar 

  • Kwiatkowski, D., P.C.B. Phillips, P. Schmidt, and Y. Shin. 1992. Testing the null of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54: 159–178.

    Article  Google Scholar 

  • Lee, J.S. 1996. On the power of stationary tests using optimal bandwidth estimates. Economics Letters 51: 131–137.

    Article  Google Scholar 

  • Lee, C.C., and P.C.B. Phillips. 1994. An ARMA prewhitened long-run variance estimator. Mimeo: Yale University.

    Google Scholar 

  • Leeb, H., and B. Potscher. 2001. The variance of an integrated process need not diverge to infinity and related results on partial sums of stationary processes. Econometric Theory 17: 671–685.

    Article  Google Scholar 

  • MacKinnon, J.G. 1994. Approximate asymptotic distribution functions for unit-root and cointegration tests. Journal of Business and Economic Statistics 12: 167–176.

    Google Scholar 

  • Müller, U. 2005. Size and power of tests for stationarity in highly autocorrelated time series. Journal of Econometrics 128: 195–213.

    Article  Google Scholar 

  • Müller, U., and G. Elliott. 2003. Tests for unit roots and the initial condition. Econometrica 71: 1269–1286.

    Article  Google Scholar 

  • Nelson, D.B. 1990. Stationarity and persistence in the GARCH (1, 1) model. Econometric Theory 6: 318–334.

    Article  Google Scholar 

  • Nelson, C.R., and C. Plosser. 1982. Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Econometrics 10: 139–162.

    Article  Google Scholar 

  • Ng, S., and P. Perron. 1995. Unit root tests in ARMA models with data dependent methods for the selection of the truncation lag. Journal of the American Statistical Association 90: 268–281.

    Article  Google Scholar 

  • Ng, S., and P. Perron. 2001. Lag length selection and the construction of unit root tests with good size and power. Econometrica 69: 1519–1554.

    Article  Google Scholar 

  • Ouliaris, S., J.Y. Park, and P.C.B. Phillips. 1989. Testing for a unit root in the presence of a maintained trend. In Advances in econometrics and modelling, ed. B. Raj. Norwell, MA: Kluwer.

    Google Scholar 

  • Park, J.Y. 1990. Testing for unit roots and cointegration by variable addition. Advances in Econometrics 8: 107–133.

    Google Scholar 

  • Park, J.Y., and P.C.B. Phillips. 1988. Statistical inference in regressions with integrated processes: Part I. Econometric Theory 4: 468–497.

    Article  Google Scholar 

  • Park, J.Y., and P.C.B. Phillips. 1989. Statistical inference in regressions with integrated processes: Part II. Econometric Theory 5: 95–131.

    Article  Google Scholar 

  • Park, J.Y., and P.C.B. Phillips. 1999. Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15: 269–298.

    Google Scholar 

  • Park, J.Y., and P.C.B. Phillips. 2000. Nonstationary binary choice. Econometrica 68: 1249–1280.

    Article  Google Scholar 

  • Park, J.Y., and P.C.B. Phillips. 2001. Nonlinear regressions with integrated time series. Econometrica 69: 1452–1498.

    Article  Google Scholar 

  • Park, J.Y., and J. Sung. 1994. Testing for unit roots in models with structural change. Econometric Theory 10: 917–936.

    Article  Google Scholar 

  • Perron, P. 1989. The great crash, the oil price shock and the unit root hypothesis. Econometrica 57: 1361–1401.

    Article  Google Scholar 

  • Phillips, P.C.B. 1986. Understanding spurious regressions in econometrics. Journal of Econometrics 33: 311–340.

    Article  Google Scholar 

  • Phillips, P.C.B. 1987a. Time series regression with a unit root. Econometrica 55: 277–301.

    Article  Google Scholar 

  • Phillips, P.C.B. 1987b. Towards a unified asymptotic theory of autoregression. Biometrika 74: 535–547.

    Article  Google Scholar 

  • Phillips, P.C.B. 1988a. Regression theory for near-integrated time series. Econometrica 56: 1021–1044.

    Article  Google Scholar 

  • Phillips, P.C.B. 1988b. Multiple regression with integrated processes. In Statistical inference from stochastic processes, ed. N.U. Prabhu. Contemporary Mathematics 80: 79–106.

    Google Scholar 

  • Phillips, P.C.B. 1991a. To criticize the critics: An objective Bayesian analysis of stochastic trends. Journal of Applied Econometrics 6: 333–364.

    Article  Google Scholar 

  • Phillips, P.C.B. 1991b. Bayesian routes and unit roots: de rebus prioribus semper est disputandum. Journal of Applied Econometrics 6: 435–474.

    Article  Google Scholar 

  • Phillips, P.C.B. 1992. The long-run Australian consumption function reexamined: An empirical exercise in Bayesian inference. In Long run equilibrium and macroeconomic modelling, ed. C. Hargreaves. Cheltenham: Edward Elgar.

    Google Scholar 

  • Phillips, P.C.B. 1994. Model determination and macroeconomic activity. Fisher–Schultz Lecture to the European Meetings of the Econometric Society, Maastricht. Discussion Paper No. 1083, Cowles Foundation, Yale University.

    Google Scholar 

  • Phillips, P.C.B. 1995. Bayesian model selection and prediction with empirical applications. Journal of Econometrics 69: 289–332.

    Article  Google Scholar 

  • Phillips, P.C.B. 1996. Econometric model determination. Econometrica 64: 763–812.

    Article  Google Scholar 

  • Phillips, P.C.B. 1998. New tools for understanding spurious regressions. Econometrica 66: 1299–1326.

    Article  Google Scholar 

  • Phillips, P.C.B. 2001. New unit root asymptotics in the presence of deterministic trends. Journal of Econometrics 11: 323–353.

    Google Scholar 

  • Phillips, P.C.B. 2006. When the tail wags the unit root limit distribution. Mimeo: Yale University.

    Google Scholar 

  • Phillips, P.C.B., and S.N. Durlauf. 1986. Multiple time series regression with integrated processes. Review of Economic Studies 53: 473–496.

    Article  Google Scholar 

  • Phillips, P.C.B., and C. Han. 2007. Gaussian inference in AR(1) time series with or without a unit root. Econometric Theory 24(3).

    Google Scholar 

  • Phillips, P.C.B., and B.E. Hansen. 1990. Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57: 99–125.

    Article  Google Scholar 

  • Phillips, P.C.B., and T. Magdalinos. 2007. Limit theory for moderate deviations from a unit root. Journal of Econometrics 136: 115–130.

    Article  Google Scholar 

  • Phillips, P.C.B., and H.R. Moon. 1999. Linear regression limit theory for nonstationary panel data. Econometrica 67: 1057–1111.

    Article  Google Scholar 

  • Phillips, P.C.B., and H.R. Moon. 2000. Nonstationary panel data analysis: an overview of some recent developments. Econometric Reviews 19: 263–286.

    Article  Google Scholar 

  • Phillips, P.C.B., and S. Ouliaris. 1990. Asymptotic properties of residual based tests for cointegration. Econometrica 58: 165–193.

    Article  Google Scholar 

  • Phillips, P.C.B., and P. Perron. 1988. Testing for a unit root in time series regression. Biometrika 75: 335–346.

    Article  Google Scholar 

  • Phillips, P.C.B., and W. Ploberger. 1994. Posterior odds testing for a unit root with data-based model selection. Econometric Theory 10: 774–808.

    Article  Google Scholar 

  • Phillips, P.C.B., and W. Ploberger. 1996. An asymptotic theory of Bayesian inference for time series. Econometrica 64: 381–413.

    Article  Google Scholar 

  • Phillips, P.C.B., and V. Solo. 1992. Asymptotics for linear processes. Annals of Statistics 20: 971–1001.

    Article  Google Scholar 

  • Phillips, P.C.B., and Z. Xiao. 1998. A primer on unit root testing. Journal of Economic Surveys 12: 423–469.

    Article  Google Scholar 

  • Phillips, P.C.B., J.Y. Park, and Y. Chang. 2004. Nonlinear instrumental variable estimation of an autoregression. Journal of Econometrics 118: 219–246.

    Article  Google Scholar 

  • Ploberger, W. 2004. A complete class of tests when the likelihood is locally asymptotically quadratic. Journal of Econometrics 118: 67–94.

    Article  Google Scholar 

  • Pötscher, B.M. 2004. Nonlinear functions and convergence to Brownian motion: Beyond the continuous mapping theorem. Econometric Theory 20: 1–22.

    Article  Google Scholar 

  • Robinson, P.M. 1995. Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23: 1630–1661.

    Article  Google Scholar 

  • Said, S.E., and D.A. Dickey. 1984. Testing for unit roots in autoregressive moving average models of unknown order. Biometrika 71: 599–608.

    Article  Google Scholar 

  • Sargan, J.D., and A. Bhargava. 1983. Testing residuals from least squares regression for being generated by the Gaussian random walk. Econometrica 51: 153–174.

    Article  Google Scholar 

  • Schmidt, P., and P.C.B. Phillips. 1992. LM tests for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54: 257–287.

    Article  Google Scholar 

  • Schotman, P., and H.K. van Dijk. 1991. A Bayesian analysis of the unit root in real exchange rates. Journal of Econometrics 49: 195–238.

    Article  Google Scholar 

  • Shimotsu, K., and P.C.B. Phillips. 2005. Exact local whittle estimation of fractional integration. Annals of Statistics 33: 1890–1933.

    Article  Google Scholar 

  • Sims, C.A., and H. Uhlig. 1991. Understanding unit rooters: A helicopter tour. Econometrica 59: 1591–1599.

    Article  Google Scholar 

  • So, B.S., and D.W. Shin. 1999. Cauchy estimators for autoregressive processes with applications to unit root tests and confidence intervals. Econometric Theory 15: 165–176.

    Article  Google Scholar 

  • Solo, V. 1984. The order of differencing in ARIMA models. Journal of the American Statistical Association 79: 916–921.

    Article  Google Scholar 

  • Stock, J.H. 1991. Confidence intervals for the largest autoregressive root in US macroeconomic time series. Journal of Monetary Economics 28: 435–459.

    Article  Google Scholar 

  • Stock, J. 1994a. Deciding between I(1) and I(0). Journal of Econometrics 63: 105–131.

    Article  Google Scholar 

  • Stock, J.H. 1994b. Unit roots, structural breaks and trends. In Handbook of econometrics, ed. R.F. Engle and D. McFadden, Vol. 4. Amsterdam: North-Holland.

    Google Scholar 

  • Stock, J.H. 1999. A class of tests for integration and cointegration. In Cointegration, causality and forecasting: A Festschrift in honour of Clive W. J. Granger, ed. R.F. Engle and H. White. Oxford: Oxford University Press.

    Google Scholar 

  • Stock, J.H., and M.W. Watson. 1988. Variable trends in economic time series. Journal of Economic Perspectives 2(3): 147–174.

    Article  Google Scholar 

  • Sul, D., P.C.B. Phillips, and C.-Y. Choi. 2006. Prewhitening bias in HAC estimation. Oxford Bulletin of Economics and Statistics 67: 517–546.

    Article  Google Scholar 

  • Wang, Q., and P.C.B. Phillips. 2006. Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Discussion Paper no. 1594, Cowles Foundation, Yale University.

    Google Scholar 

  • Wei, C.Z. 1992. On predictive least squares principles. Annals of Statistics 20: 1–42.

    Article  Google Scholar 

  • Xiao, Z., and P.C.B. Phillips. 1998. An ADF coefficient test for a unit root in ARMA models of unknown order with empirical applications to the U.S. economy. Econometrics Journal 1: 27–43.

    Article  Google Scholar 

  • Zivot, E., and D.W.K. Andrews. 1992. Further evidence on the great crash, the oil price shock and the unit root hypothesis. Journal of Business and Economic Statistics 10: 251–270.

    Google Scholar 

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    Peter C. B. Phillips

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Phillips, P.C.B. (2018). Unit Roots. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95189-5_2586

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